 When we evaluate a double integral over a region R, we could evaluate either the double integral first with respect to x and then y, or the double integral first with respect to y and then x, depending on whether we sum by allowing x or y to vary. To avoid commitment until we're ready, we use the differential of area, written DA, and write our double integral this way. But remember, to compute, you have to commit. For example, let R be the region bounded by the x and y axes since the line x plus 3y equals 12, and let's write two possible computational forms of this double integral. So let's graph our region. Now, if we integrate with respect to x first, that's letting our x values vary, and so we're moving horizontally. And so our x values will run from the x axis to the line x plus 3y equals 12. Now, the important thing here to remember is the differential variable is controlling, and since we're integrating with respect to x first, our limits of integration have to be x equals something to x equals something. Since we're starting at the x axis, our lower bound is going to be x equals zero. Since we're going to the line x plus 3y equals 12, we'll rewrite that so it's x equals something, and that gives us our upper and lower bounds. Now, our y value will then integrate with respect to y, and so that means we'll move vertically. Now, our y values seem to run from the x axis to the line x plus 3y equals 12. But, since we've already integrated with respect to x, our limits can't include x. Remember, once you've integrated the variable vanishes. Instead, we need to find the highest and lowest points in the region. So now we see our y values run from y equals zero to y equals four, and that gives us the limits of integration with respect to y. We can also try to integrate this the other way, so if we integrate with respect to y first, we see that our y values, that's our vertical distance, run from the x axis to the line x plus 3y equals 12. And again, remember the differential variable is controlling, so for integrating with respect to y, then our limits are going to go from y equals something to y equals something. The x axis has equation y equals zero, and we have the equation of the line, so we'll solve this for y and get. And next, we want to integrate with respect to x, and so we'll move left and right, and so our x values run from x equals zero all the way up until we get x equals 12. Now, because we have a choice of directions, we should always use whichever direction allows us to find the integral. For example, let's consider the region R bound by the lines y equals 1, y equals e to the x, and x equals 2, and let's set up two integrals to evaluate the area of the region, and then pick one of them and evaluate. So we'll graph the region, and since we know we'll probably need them, we'll go ahead and locate these intersection points as well. So if we integrate with respect to x first, let's see what we get. If we let x vary, x goes from the graph of y equals e to the x to x equals 2. But again, the differential variable is controlling, so we need to express both of these in terms of x equals something. So we want to solve y equals e to the x for x, and we get, and so our limits of integration will run from x equals log y to x equals 2. Now we let y vary, and we see that y goes from the x-axis up to y equals e to the x, but remember, once you've integrated, the variable vanishes. Since we've already integrated with respect to x, our limits can't contain x. Instead, we see that our lowest y values are at y equals 1, and our highest are at y equals e squared. And so y varies from y equals 1 to y equals e squared. Now let's see what would happen if we tried to integrate this as written. If we integrate with respect to x first, we first evaluate, but then we need to evaluate the integral involving a log. And maybe you're one of those people who look at this and say, integrating a log, that's what I've wanted to do my entire life. My life would be unfulfilled if I couldn't integrate log. As for me, I know there's a formula for it, but I always forget what it is. So let's see if we can integrate with respect to y first. Now the important thing here is that we can't just switch our limits of integration, because again, once we've integrated with respect to a variable, we can't use it anymore. So we've got to go back to our graph and redetermine those limits of integration. So if we're integrating with respect to y first, we move vertically, and so if we let y vary, then y goes from y equals 1 to the graph of y equals e to the x. So these will be our limits of integration. Then x varies from x equals 0 all the way up to x equals 2. And that allows us to find the value of the definite integral. So we'll integrate with respect to y first. And since our integrand is e to the x, this is the world's easiest antiderivative, we can find the second integral, which gives us the area of the region.